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THE TEACHING OF MATHEMATICS IN PREPARATORY SCHOOLS.*

The study of mathematics in preparatory schools, though obviously not extensive, is nevertheless of the utmost importance. Limit in the number of subjects, limit also in the range of these subjects, there must necessarily be; a limit easily ascertained when the proportion of time that may be fairly devoted to mathematics and when the thinking capabilities of an average boy of 12 to 14 years are fully considered. Assuming that the days of specialisation are gone for ever, assuming also that Preparatory" is strictly interpreted to mean "under 14," the range of study is "cribbed, cabined, and confined" within very narrow bounds.

Our considerations will naturally fall under two heads :

(a). Preparation for public school entrance.

(b). Preparation for public school scholarships.

And yet it must not be concluded that these heads represent two distinct branches of education; for all practical purposes of teaching they go hand in hand. No preparatory school master, who aims at sound work, makes any distinction between possible candidates for scholarship and the ordinary rank and file as represented by the average boy. Though the former will always outdistance the latter, yet the process of education must always remain the same, the only tangible difference being that the one is capable of a more extended course than the other, and this difference is fully provided for in the more advanced work of the higher classes to which the average boy rarely or ever attains.

The curriculum of a preparatory school is nothing if not sufficiently clastic to admit of a different classification of boys according to individual attainments and capabilities in cach individual subject. Thus the same boy may be in one set for classics, in another for French, in yet another for mathematics; this is a fact that must be fully grasped in any study of English secondary schools. Under any other form of classification a boy will be almost certainly taking one of two courses; either he will be doing work which is insufficient for his requirements, which means losing time, or he will be going too far ahead, in which case he will inevitably become inaccurate and unsound. It is quite clear that an independent classification for each individual subject is of the greatest advantage both to masters and boys: to the former in providing them with a class as level as possible in knowledge and powers, to the latter in affording means of steady uniform progress in every subject that is required of them.

In the lower forms boys, whether their goal be entrance or scholarship, will naturally work together, the more clever boys being slightly younger than the rest of the class. And this

*We much regret that Mr. Allum died as this paper was passing through the press, and that it has not therefore had the advantage of receiving his final corrections.-ED.

system will continue all through the school, so that by some law of gravitation the average boy will not rise either so quickly or so high as his more gifted schoolfellow; it is therefore this fact alone rather than the wishes of the parent or the aim of the boy that eventually will decide whether a boy will have a reasonable chance of a scholarship. In a well-organized preparatory school the boys of the highest form reach the standard of public school scholarships by the time they attain the limits of age (12-14), and it is quite certain that, if any alteration is made in the length of the working day, it is in the direction of curtailment rather than in that of extension. A fresh brain is capable of more good work than one that is fatigued and dulled by a long period of hard exertion. The brain must have rest in order to grow, while a long period of severe strain would probably retard the growing process to such a degree that the brain power of what might under other conditions have been a forward boy of 14 is little, if any, more than it was two years before.

Scholarship classes, as apart from highest forms, are perfectly unnecessary and harmful-the wheat and the tares must grow together all through the school; the weaker boys will be left behind only by their inability to acquire knowledge as quickly as their other contemporaries.

By no means also let there be any specialisation of subjects to the neglect of others. True education is an impartial and, as far as possible, an equal development of all faculties in due proportion. It is quite true that some young boys show special taste for classics, others an aptitude for mathematics, yet better educational results are obtained-by which I mean more thinking power-by a judicious latitude of curriculum, than by devoting a preponderance of time and effort to the exclusive development of any individual study.

In the case of young boys mathematical genius is by nature limited, and though it is far more conspicuous in the case of some than of others, yet there will be no perceptible retardation of the mathematical power latent in the individual, if work which is more advanced than the juvenile mind should be permitted to attempt be deferred to years of greater discretion. Of course, this by no means precludes the extension of the usual limits in the case of a boy with a more than average taste for mathematics; provided only that the time devoted to the subject be not extended, good results only can accrue from more advanced work in the case of one able to receive it. Special ability for classics or for mathematics can be met by special credit in the form of marks in the weekly, monthly, or terminal totals.

At the present time the public schools that offer scholarships for special subjects, e.g., classics or mathematics, may be counted on the fingers of one hand, so that there is a large and important consensus of opinion on the part of public school headmasters, which should go far to strengthen the hands of preparatory school headmasters in offering the most stringent opposition to specialisation. Does the principle of specialisation produce a

better selection of scholars? I think not; a glance at the honours obtained by the public schools at the universities does not encourage this opinion-rather the reverse. The selection of a scholarship roll by aggregate of marks obtained in all subjects will invariably contain the most able boys, and therefore those most likely to succeed in a future career of honours.

Granted then (i.) that special scholarship classes are unnecessary; (ii) that exclusive training in one subject is harmful; (iii.) that long hours of study defeat their object; the question arises: What proportion of time can fairly, and with advantage, be devoted to mathematics in the preparatory school curriculum? It is generally agreed that a daily lesson of 50 minutes or of one hour, according to the subdivision of hours in the school, is essential. Four of these lessons are to be devoted to analytical work, eg. arithmetic and algebra; the remaining two then fall to geometry, e.g., Euclid. In the case of the younger boys not yet able to attack Euclid, a daily lesson of arithmetic, especially including mental calculation, will soon bring about the time for attacking algebra and Euclid.

ARITHMETIC.

It may be safely assumed that on entrance at a preparatory school every boy is acquainted with the simple and compound rules. The work of the preparatory school may be understood to commence from this point. It is a fruitful cause of delay to waste much time over numeration and notation; for all practical purposes of the beginner it is useless to go beyond seven figures-in fact, hundreds and thousands suffice for most elementary work. There is nothing to be gained by very long sums; as a matter of fact they discourage by fatiguing small minds; better results in method and accuracy are to be attained by limiting the number of figures in a multiplicand or quotient to five or six than by courting inaccuracy by lengthy processes.

Now is the time to attack easy arithmetical problems, and the more these are adopted the better arithmeticians the pupils become. The sooner the notion that sums are to be worked by rule is dispelled the better; it is intelligence, not memory, that is to work the oracle.

There is no better training than the solution of easy questions by unitary method, and at this stage it should be thoroughly taught. Let all rule of thumb methods of so-called "rule of three" be once and for ever discarded, and let the pupil be taught to reason for himself from each question by reduction to unity, and there will be a manifest gain. After a course of fractions the questions can be made more difficult, but on the other hand their solution will be the easier by the knowledge of the use of fractions.

Factors must be utilised to the fullest extent, both in multiplication and division, and in the latter the true remainder should always be found. Facility in finding factors should be encouraged both as a means of shortening calculations, and as a

development of a quick power to perceive the constituents of a number. It is this same readiness which later produces the power to attack problems both in arithmetic and algebra.

A simple method easily caught by even young boys is as follows:--

What are the factors of 144 and of 999 ?

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It is easily seen that the divisors are here found in order 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144, while for purposes of multiplication and division the pair most suitable can be selected.

G.C.M. and L.C.M. should be, whenever practicable, worked by factors, and it should be clearly impressed that cancelling in fractional sums is simply division by the G.C.M.

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Vulgar Fractions.-Fractions are frequently deferred too long. As soon as a child can grasp the nature of 1. and d., which he is ready to do very quickly from the fact of a tangible picture being conveyed to his mind, he should be taught to add, or subtract, other fractions such as 1, 3, 3, 3, 3, 1, 1, 8, §, 1, &c. and then he may easily proceed to easy fractions whose denominators lie in the same table of multiplication as,,,, &c. A simple geometrical figure will soon show him that == ;= 4, &c., and he will readily adapt this to other numbers. In this way a

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valuable intuition into the nature of fractions is obtained, which has time to be thoroughly digested, and therefore infallibly grasped as a preparation for the later stage of unfathoming the mysteries of vulgar fractions. Nor will the mysteries be long undiscovered, for any child of intelligence, to whom the nature of a fraction has been properly explained, will dispose with socalled rules and work out his results by the light of his intelligence. It is important to emphasise that the simplification of compound and complex fractions must be uniformly progressive, that is to say each part must be advanced one stage in each successive line, until all are alike homogeneous, either in terms or factors.

N.B. The whole question must be set down and attacked at once; it is bad method to work by instalments; whenever possible, the sign of equality should be in the middle of the paper, the question on the left, and each successive stage of the solution on the right in column. It is also of importance that all calculation that cannot be made in the head should be shown on the actual paper; rough work on stray papers should never be allowed. For teaching purposes the rough work is equally valuable with

the final results, for it gives the teacher an insight into the course of reasoning that the pupil has adopted.

Decimal Fractions.-A good explanation of the law that governs our general scale of notation will simplify matters considerably, and in few cases will there be any difficulty in realising that the same decimal system that is used in the formation of whole numbers is naturally and simply extended below unity to represent fractions. Taking the units figure as the starting point, it is at once seen that tens and tenths, and hundreds and hundredths, &c., run in pairs, equidistant on either side of the units figure, the decimal point marking the division between whole numbers and fractions. The importance of local value in a clear understanding of decimal fractions cannot be too strongly urged. If once thoroughly made clear, the decimal point, instead of being, as it too frequently is made, a bête noir, is a veritable friend, and any difficulty in division is once and for ever dispelled.

Interest, discount, percentages, profit and loss, stocks, and all the host of so-called rules, (why "rules"?) are completely brought within the reach of an average intelligence by a thorough explanation of the definitions that give rise to the names, and it is not only needless, but destructive of thinking power to teach these as hard and fast rules. The application of reasoning by unitary method to the definitions will always provide the shortest and the easiest method of ascertaining the answers, one too that must be understood because it is the outcome of a logical train of thought. One need not emphasise this, because it will be manifest to all teachers that whatever can be attained by reasoning faculties must be indelibly fixed on the mind, while all that is acquired by memory will just as inevitably be an unreliable quantity.

There remain only problems of time and work, and those cannot be classified under a definite name. These are of great value, as inducing independent thought; and some slight knowledge of the relation between units of velocity, time and space will be required. This, as a matter of fact, presents no difficulty, and is easily acquired by a short blackboard demonstration.

As a general axiom it may be assumed that it is not possible to dispense with blackboard teaching. In fact the more the blackboard is used the better, and one can almost estimate the value of a teacher by the quantity of chalk he uses. Nor must the work be done entirely by the teacher; each pupil should be encouraged to do the successive lines of work, sometimes entirely, sometimes in turn with others, and it is a good plan after a demonstration to call upon one or more members of the form to reproduce it on the board so as to ensure a through grasp of the problem.

ALGEBRA.

The quantity of algebra that may be attempted with advantage by boys of preparatory age is not quite agreed upon. Some would wish to include indices, surds, and everything to quadratic equations (inclusive); others are of opinion that it is

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